Estimating model parameters for oil and gas exploration from geophysical data is challenging and subject to a large degree of uncertainty. Seismic imaging techniques, such as seismic amplitude versus angle (AVA) and amplitude versus offset (AVO) inversion, can produce highly accurate estimates of the physical location and porosity of potential reservoir rocks, but in many circumstances has only a limited ability to discriminate the fluids within the reservoir. Other geophysical data such as electromagnetic (EM) methods can add information about water saturation, and by extension hydrocarbon saturations, because the electrical conductivity of rocks is highly sensitive to water saturation. However, estimating fluid saturation using EM data alone is impractical because EM data have low spatial resolution. Seismic and EM methods are sensitive to different physical properties of reservoir materials: seismic data are functions of the seismic P- and S-wave velocity and density of the reservoir, and EM data are functions of the electrical resistivity of the reservoir. Because both elastic and electrical properties of rocks are related physically to fluid saturation and porosity through rock-physics models, joint inversion of multiple geophysical data sets such as seismic data and EM data has the potential to provide better estimates of earth model parameters such as fluid saturation and porosity than inversion of individual data sets.
Prior art inversion of geophysical data to derive estimates of model error and model parameters commonly relies on gradient based techniques which minimize an object function that incorporates a data misfit term and possibly an additional model regularization or smoothing term. For example, Equation (1) is a general object function, φ, commonly usedφ(m, d)=[D(do−dp)]H[(D(dobs−dp))]+λ(Wm)H(Wm)  (1)D is the data covariance matrix, do and dp are the observed and predicted data respectively, W is the model regularization matrix, m is the vector of model parameters, this could be electrical conductivity, and λ is the trade-off parameter that scales the importance of model smoothing relative to data misfit. H denotes the transpose-conjugation operator since the data d is complex. Linearizing equation (1) about a given model, mi, at the ith iteration produces the quadratic form(JTSTSJ+λWTW)mi+1=JTSTSJmi+JTSTSδdi  (2)where mi+1 can be solved for using many techniques, a quadratic programming algorithm is one possibility. J is the Jacobian matrix of partial derivatives of data with respect to model parameters, S is the matrix containing the reciprocals of the data's standard deviations, such that ST=D−1. The current difference between calculated (dp) and observed (dobs) data is given by δdi=dobs−di. The trade-off parameter λ is adjusted from large to small as iterations proceed.
When the algorithm, described by equations (1) and (2), converges to a minimum of the object function, φ, a single model, m is produced. This prior art derived model is not guaranteed in any way to be the “global” or true model. Model parameter error, also known as model parameter standard deviations (the square root of the variance), estimates derived from model parameter covariance calculations, such as described by equations (1) and (2), are not accurate and provide an insufficient quantification of the true model parameter errors.
Unlike prior art inverse methods, inversion of geophysical data sets using a sampling-based stochastic model can provide an accurate estimate of the probability density functions (PDF's) of all model parameter values. Further, the sampling-based stochastic method can be used for joint inversion of multiple geophysical data sets, such as seismic and EM data, for better estimates of earth model parameters than inversion of individual data sets. The term stochastic inversion is used widely to cover many different approaches for determining the PDF's of model parameter variables. The model parameter PDF's provide an accurate estimate of the variance of each model parameter and the mean, mode and median of the individual model parameters. The accurate model parameter variances can be used when comparing multiple models to determine the most probable model for an accurate interpretation of the earth's subsurface.